Subgroups — C₃₁×#22822 #21479581

Order 7936.

656 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2295	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1917	C₂	1	2	yes	maximal
5	C₃₁×#1916	C₂	1	2	yes	maximal
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1832	C₂	1	2	yes	maximal
8	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1768	C₂	1	2	yes	maximal
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#391	C₄	1	1	yes	characteristic
24	C₃₁×#391	C₄	1	1	yes	characteristic
25	C₃₁×#389	C₄	1	1	yes	characteristic
26	C₃₁×#389	C₄	1	1	yes	characteristic
27	C₃₁×#387	C₄	1	2	yes	—
28	C₃₁×#387	C₄	1	2	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#376	C₂²	1	2	yes	—
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊C₈	—	2	2	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	4	no	—
69	C₃₁×C₄⋊C₈	—	4	2	no	—
70	C₃₁×C₄⋊C₈	Q₈	1	4	yes	complemented
71	C₃₁×#97	—	4	2	no	—
72	C₄×C₈×C₃₁	—	2	2	no	—
73	C₄×C₈×C₃₁	—	2	2	no	—
74	C₄×C₈×C₃₁	—	4	2	no	—
75	C₃₁×#95	C₂×C₄	1	2	yes	—
76	C₃₁×#95	C₂×C₄	1	2	yes	—
77	C₃₁×#95	C₂×C₄	1	2	yes	—
78	C₃₁×#95	C₂×C₄	1	2	yes	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₂×C₈×C₃₁	—	2	1	no	—
97	C₂×C₈×C₃₁	—	2	1	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	1	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	1	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄²×C₃₁	—	2	2	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	—	2	1	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	4	1	no	—
130	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
131	C₄²×C₃₁	#41	1	2	yes	—
132	C₄²×C₃₁	#41	1	2	yes	—
133	C₄²×C₃₁	#41	1	2	yes	—
134	#22822	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
136	Q₈×C₃₁	—	4	4	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	#117	1	2	yes	—
161	C₂×C₄×C₃₁	#117	1	2	yes	—
162	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
163	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
164	C₂×C₄×C₃₁	#131	1	2	yes	—
165	C₂×C₄×C₃₁	#131	1	2	yes	—
166	C₂×C₄×C₃₁	#131	1	2	yes	—
167	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
173	C₈×C₃₁	—	2	2	no	—
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	4	1	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	2	no	—
179	#2295	C₂×C₃₁	1	1	yes	characteristic
180	#1984	C₂×C₃₁	1	1	yes	characteristic
181	#1984	C₂×C₃₁	1	1	yes	characteristic
182	#1917	C₂×C₃₁	1	2	yes	—
183	#1916	C₂×C₃₁	1	2	yes	—
184	#1842	C₂×C₃₁	1	1	yes	characteristic
185	#1832	C₂×C₃₁	1	2	yes	—
186	#1770	C₂×C₃₁	1	1	yes	characteristic
187	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
188	#1768	C₂×C₃₁	1	2	yes	—
189	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
190	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
191	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₂²×C₃₁	#520	1	1	yes	characteristic
193	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
194	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
195	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
196	C₂²×C₃₁	#568	1	1	yes	characteristic
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	4	2	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	—	4	1	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	#433	1	4	yes	—
207	C₄×C₃₁	#554	1	1	yes	characteristic
208	C₄×C₃₁	#556	1	1	yes	characteristic
209	C₄×C₃₁	#557	1	1	yes	characteristic
210	C₄×C₃₁	#559	1	1	yes	characteristic
211	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#97	C₂²×C₃₁	1	2	yes	—
219	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
222	#391	C₄×C₃₁	1	1	yes	characteristic
223	#391	C₄×C₃₁	1	1	yes	characteristic
224	#389	C₄×C₃₁	1	1	yes	characteristic
225	#389	C₄×C₃₁	1	1	yes	characteristic
226	#387	C₄×C₃₁	1	2	yes	—
227	#387	C₄×C₃₁	1	2	yes	—
228	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
230	#376	C₂²×C₃₁	1	2	yes	—
231	C₄³	C₂²×C₃₁	1	1	yes	characteristic
232	#336	C₂²×C₃₁	1	2	yes	—
233	#336	C₂²×C₃₁	1	2	yes	—
234	#336	C₂²×C₃₁	1	2	yes	—
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	1	yes	characteristic
237	#336	C₂²×C₃₁	1	2	yes	—
238	#336	C₂²×C₃₁	1	1	yes	characteristic
239	#336	C₂²×C₃₁	1	2	yes	—
240	C₂×C₃₁	#2355	1	1	yes	characteristic
241	C₂×C₃₁	C₂×#433	1	1	yes	characteristic
242	C₂×C₃₁	#2925	1	1	yes	characteristic
243	C₂×C₃₁	#2960	1	1	yes	characteristic
244	C₂×C₃₁	#2962	1	1	yes	characteristic
245	C₂×C₃₁	#2993	1	1	yes	characteristic
246	C₂×C₃₁	#2994	1	1	yes	characteristic
247	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
248	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
253	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
254	C₄×Q₈	—	2	2	no	—
255	C₄×Q₈	—	2	2	no	—
256	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	2	2	no	—
274	C₄⋊C₈	—	2	4	no	—
275	C₄⋊C₈	—	4	2	no	—
276	C₄⋊C₈	Q₈×C₃₁	1	4	yes	complemented
277	#97	—	4	2	no	—
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	2	2	no	—
280	C₄×C₈	—	4	2	no	—
281	#95	C₂×C₄×C₃₁	1	2	yes	—
282	#95	C₂×C₄×C₃₁	1	2	yes	—
283	#95	C₂×C₄×C₃₁	1	2	yes	—
284	#95	C₂×C₄×C₃₁	1	2	yes	—
285	#95	C₂×C₄×C₃₁	1	2	yes	—
286	#95	C₂×C₄×C₃₁	1	2	yes	—
287	#95	C₂×C₄×C₃₁	1	2	yes	—
288	C₃₁	#22822	1	1	yes	characteristic, direct product factor, Sylow
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
295	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
297	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
298	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
299	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
320	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	4	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	1	no	—
337	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
338	C₄²	#41×C₃₁	1	2	yes	—
339	C₄²	#41×C₃₁	1	2	yes	—
340	C₄²	#41×C₃₁	1	2	yes	—
341	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	C₃₁×#117	1	2	yes	—
367	C₂×C₄	C₃₁×#117	1	2	yes	—
368	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
369	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
370	C₂×C₄	C₃₁×#131	1	2	yes	—
371	C₂×C₄	C₃₁×#131	1	2	yes	—
372	C₂×C₄	C₃₁×#131	1	2	yes	—
373	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
377	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
378	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
379	C₈	—	2	2	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	1	no	—
383	C₈	—	4	2	no	—
384	C₈	—	4	2	no	—
385	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
386	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
387	C₂²	C₃₁×#520	1	1	yes	characteristic
388	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
389	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
390	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
391	C₂²	C₃₁×#568	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	2	2	no	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	2	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	C₃₁×#433	1	4	yes	—
402	C₄	C₃₁×#554	1	1	yes	characteristic
403	C₄	C₃₁×#556	1	1	yes	characteristic
404	C₄	C₃₁×#557	1	1	yes	characteristic
405	C₄	C₃₁×#559	1	1	yes	characteristic
406	C₂	C₃₁×#2355	1	1	yes	characteristic
407	C₂	C₂×C₃₁×#433	1	1	yes	characteristic
408	C₂	C₃₁×#2925	1	1	yes	characteristic
409	C₂	C₃₁×#2960	1	1	yes	characteristic
410	C₂	C₃₁×#2962	1	1	yes	characteristic
411	C₂	C₃₁×#2993	1	1	yes	characteristic
412	C₂	C₃₁×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
70. C₃₁×C₄⋊C₈	Q₈	semidirect
134. #22822	C₃₁	direct (N × H)
276. C₄⋊C₈	Q₈×C₃₁	semidirect
288. C₃₁	#22822	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2295	2
C₃₁×#1984	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1842	2
C₃₁×#1832	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1768	2
C₄×C₃₁×C₄⋊C₈	2
#22822	31

Subgroups — C31×#22822 #21479581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22822 #21479581